In this case, it is the mean of the difference between the two values that is distributed according to the t distribution.
This example is from Dalgard.
Not only are the values correlated, but the difference is always negative:
We can do the test by hand, as follows:
The question now is, what fraction of the values from the t-distribution with df = 10 are greater than 11.39?
The short answer: not very many!